Let $\bf{n}'$ be a unit vector in the direction of a wavevector in the plasma rest frame and $\bf{B'}$ be a unit vector along the magnetic field in the plasma rest frame. The electric field of a linearly polarized electromagnetic wave is directed along the unit vector $\hat{e}=\bf{n'} \times \hat{B'}$, and the magnetic field of the wave is along the unit vector $\hat{b'}=\bf{b}\times \hat{e'}$, such that the Poynting flux along is $\hat{e'}\times \hat{b'}$ directed along $\bf{n'}$. We give a Lorentz boost to the explosion frame to find the electric field $e$ there, normalize it to unity, and project e$$ on some given direction (e.g. , along the projection of the flow axis on the plane of the sky).
Fields in the wave expressed in terms of the direction of a photon in the explosion frame $\bf{n}$ is
\begin{equation} \hat{e'} ~=~\frac{\bf{b} \times \hat{B'}}{\Gamma (1-\bf{n}\cdot\bf{v})}+\frac{1+\Gamma (1-\bf{n}\cdot \bf{v})}{(1+\Gamma(1-\bf{n}\cdot \bf{v}))} \end{equation}
where $\Gamma=\frac{1}{\sqrt{1-v^2/c^2}}$ is the bulk Lorentz factor. Would some one help to deduce the above equation?
