Editing Waveplate (section)

=== Half-wave plate ===
[[File:Waveplate notext.png|thumb|right|A wave passing through a half-wave plate]]
For a half-wave plate, the relationship between ''L'', Δ''n'', and λ<sub>0</sub> is chosen so that the phase shift between polarization components is Γ&nbsp;= π. Now suppose a linearly polarized wave with polarization vector <math>\mathbf{\hat p}</math> is incident on the crystal. Let θ denote the angle between <math>\mathbf{\hat p}</math> and <math>\mathbf{\hat f}</math>, where <math>\mathbf{\hat f}</math> is the vector along the waveplate's fast axis. Let ''z'' denote the propagation axis of the wave. The electric field of the incident wave is
<math display="block">\mathbf{E}\,\mathrm{e}^{i(kz-\omega t)} = E\, \mathbf{\hat p}\,\mathrm{e}^{i(kz-\omega t)} = E (\cos\theta\, \mathbf{\hat f} + \sin\theta\, \mathbf{\hat s}) \mathrm{e}^{i(kz-\omega t)},</math>
where <math>\mathbf{\hat s}</math> lies along the waveplate's slow axis. The effect of the half-wave plate is to introduce a phase shift term e<sup>''i''Γ</sup>&nbsp;= e<sup>''i''π</sup>&nbsp;= −1 between the ''f'' and ''s'' components of the wave, so that upon exiting the crystal the wave is now given by
<math display="block">E (\cos\theta\, \mathbf{\hat f} - \sin\theta\, \mathbf{\hat s}) \mathrm{e}^{i(kz-\omega t)} = E [\cos(-\theta) \mathbf{\hat f} + \sin(-\theta) \mathbf{\hat s}] \mathrm{e}^{i(kz-\omega t)}.</math>
If <math>\mathbf{\hat p}'</math> denotes the polarization vector of the wave exiting the waveplate, then this expression shows that the angle between <math>\mathbf{\hat p}'</math> and <math>\mathbf{\hat f}</math> is −θ. Evidently, the effect of the half-wave plate is to mirror the wave's polarization vector through the plane formed by the vectors <math>\mathbf{\hat f}</math> and <math>\mathbf{\hat z}</math>. For linearly polarized light, this is equivalent to saying that the effect of the half-wave plate is to rotate the polarization vector through an angle 2θ; however, for elliptically polarized light the half-wave plate also has the effect of inverting the light's [[chirality|handedness]].<ref name="hecht" />