 Let "z" denote the propagation axis of the wave.The electric field of the incident wave is
where formula_7 lies along the waveplate's slow axis.The effect of the half-wave plate is to introduce a phase shift term e"i"Γ = e"i"π = −1 between the "f" and "s" components of the wave, so that upon exiting the crystal the wave is now given by
If formula_9 denotes the polarization vector of the wave exiting the waveplate, then this expression shows that the angle between formula_9 and formula_4 is −θ.