  Ultra-high energy cosmic rays can be measured through the detection of
radio-frequency radiation from air showers. The radio-frequency emission
originates from deflections of the air-shower particles in the geomagnetic
field and from a time-varying negative charge excess in the shower front. The
distribution of the radio signal on the ground contains information on crucial
cosmic-ray properties, such as energy and mass. A long standing challenge is to
access this information experimentally with a sparse grid of antennas. We
present a new analytic model of the radio signal distribution that depends only
on the definition of the shower axis and on the parameters energy and distance
to the emission region. The distance to the emission region has a direct
relation to the cosmic ray's mass. This new analytic model describes the
different polarizations of the radiation and therefore allows the use of
independently measured signals in different polarization, thereby doubling the
amount of information that is available in current radio arrays, compared to
what has been used thus far. We show with the use of CoREAS Monte Carlo
simulation that fitting the measurements with our model does not result in
significant contributions in both systematic bias and in resolution for the
extracted parameters energy and distance to emission region, when compared to
the expected experimental measurement uncertainties.
