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\begin{document}

\title{Bremsstrahlung Electron Crossing Different Mediums: Extension of
Asymmetric Backward Peaking Radiation Pattern from a Relativistic
Particle Accelerated by Lightning Leader Tip Electric Field}



\author[1]{Mert Yucemoz}%
\affil[1]{University of Bath}%


\vspace{-1em}



  
  \date{June 13, 2024}


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\begin{abstract}
Previously the radiation patterns of combined parallel and perpendicular
motions from the accelerated relativistic particle at low and high
frequencies of the bremsstrahlung process with an external lightning
electric field were explained. The primary outcome was 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 forwarding and backward peaking lobes, is
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. Previously
stated bremsstrahlung asymmetry, R was an asymmetry in the radiation
lobe pairs about particles velocity vector. Bremsstrahlung asymmetry
used to occur at the same level in both forward radiation lobe pairs and
backward radiation lobe pairs. However, in high-density mediums where
the emitted wave can lag behind the speed of the particle, symmetry of
the magnitude of bremsstrahlung asymmetry, R differs between forward
peaking radiation lobe pairs relative to backward peaking radiation lobe
pairs. This is another novel asymmetry and it causes bremsstrahlung
asymmetry, R to be larger in the forward peaking compared to backward
peaking radiation. The outcome is the shrink in radiation length that
occurs in the backward peaking lobes. This extended work reports,
changes in the radiation pattern as the emitted wave propagates through
different mediums. Two novel formulas are derived from Snell's law for a
particle entering the medium horizontally and from any other angle
between \textbackslash{}Pi/2 and -\textbackslash{}Pi/2 radians. The
novel outcome is the change in angle between forward peaking radiation
lobe pair and backward peaking radiation lobe pair defined as
bremsstrahlung angle, \textbackslash{}theta\_\{brem\}. When the
bremsstrahlung particle crosses different mediums, change in angle
between the forward and backward radiation lobe pairs, bremsstrahlung
angle, \textbackslash{}theta\_\{brem\} breaks into its components as
each lobe changes angle at different magnitudes from the particle's
velocity vector. Therefore, bremsstrahlung angle,
\textbackslash{}theta\_\{brem\} between forward-backward peaking lobes
transforms into individual angles \textbackslash{}Omega\_\{1\},
\textbackslash{}Omega\_\{2\}, \textbackslash{}Omega\_\{3\}.
\textbackslash{}Omega\_\{4\} all measuring from the particle's velocity
vector.%
\end{abstract}%



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