Terrestrial Gamma-ray Flashes exhibit slopes of ionizing radiation
associated with bremsstrahlung. Bremsstrahlung has a continuous spectrum
of radiation from radio waves to ionizing radiation. The Poynting vector
of the emitted radiation, i.e., the radiation pattern around a single
particle under the external lightning electric field during interaction
with other particles or atoms, is not quite well known. The overall
radiation pattern arises from the combination of radiation of parallel
and perpendicular motions of a particle caused by the acceleration from
the lightning electric field and the bremsstrahlung. The calculations
and displays of radiation patterns are generally limited to a
low-frequency approximation for radio waves and separate parallel and
perpendicular motions. Here we report the radiation patterns of combined
parallel and perpendicular motions from accelerated relativistic
particles at low and high frequencies of the bremsstrahlung process with
an external lightning electric field. The primary outcome is that
radiation patterns have four relative maxima with two forward peaking
and two backward peaking lobes. The asymmetry of the radiation pattern,
i.e., the different intensities of forward and backward peaking lobes,
are caused by the Doppler effect. A novel outcome is that bremsstrahlung
has an asymmetry of the four maxima around the velocity vector caused by
the curvature of the particle's trajectory as it emits radiation. In
addition, change in kinetic energy of bremsstrahlung electron and
shrinking radiation lobe due to bremsstrahlung asymmetry were found to
increase electron's energy concentration towards the outer regions of
the curved trajectory. This mathematical modeling helps to better
understand the physical processes of a single particle's radiation
pattern, which might assist the interpretation of observations with
networks of radio receivers and arrays of gamma-ray detectors.