Polarization
November 29th, 2020
The purpose of this is to discuss possible polarizations of EM waves.
Linear Polarization
Polarizations are defined using one field: the E-field.
Linear Polarizer Example
Other Polarization States
For linear polarization, we say that \(\phi \equiv \phi_{x} - \phi_{y} = 0\). This is not always the case, however.
In fact, a different type of polarization, circular polarization, requires the relative phase (\(\phi\)) abides to the following: \(\phi \equiv \phi_{x} - \phi_{y} = \pm \frac{\pi}{2}\). In the scenario that the \(\pm \frac{\pi}{2}\) is positive, it can be observed that the x component of the electric field would be proportional to the \(cos\), whereas the y component would be proportional to the \(sin\). Otherwise, if \(\pm \frac{\pi}{2}\) was negative, the x component of the electric field would be proportional to the \(sin\), whereas the y component would be proportional to the \(cos\).
Circular Polarization
Two types:
- Right-handed circular polarization
- Left-handed circular polarization
Assuming a wave is going in the positive Z direction:
- Right-handed circular polarization is when the \(\phi\) component (\(\frac{\pi}{2}\)) is positive
- Left-handed circular polarization is when the \(\phi\) component (\(\frac{\pi}{2}\)) is negative
The terminology for these makes sense since for the right-handed circular polarization, the E-field goes up and to the right (since \(cos\) corresponds to \(E_{x}\)), whereas for the left-handed circular polarization, the E-field goes down and to the left (since \(cos\) corresponds to \(E_{y}\)).
Birefringence
Birefringent materials can change an individual component’s speed. Therefore, if a wave entered the material in phase, it can exit the material with the components out of phase. Thus, we can produce a circular polarized wave.
To determine the handedness of a wave after passing through a birefringent material, simply curl your fingers from the “slow” component to the “fast” component. If your thumb points in the direction of the wave propagation, then it is right-handed circular polarization, otherwise it is left-handed circular polarization.
Formulas
E-field defined using a unit vector \(\hat{e}\) which defines the direction of polarization. \(\phi\) specifies E-field strength at \(z=t=0\)
\[ E = \hat{e} E_{0} sin ( kz - \omega t + \phi ) \]
X and Y component magnitudes, where \(\theta\) is the angle between the unit vector \(\hat{e}\) and x axis.
\[ E_{x} = E_{0} cos \theta sin ( kz - \omega t + \phi ) \]
\[ E_{y} = E_{0} sin \theta sin ( kz - \omega t + \phi ) \]
Polarizers
Final Intensity after unpolarized light through polarizer
\[ I_{final} = \frac{1}{2} I_{0} \]
Final Intensity after polarized light through polarizer
\[ I_{final} = I_{0} cos^{2} \theta \]
Birefringent Phase Difference between components
\[ \Delta \phi = \phi_{y} - \phi_{x} = \omega d ( \frac{1}{v_{fast}} - \frac{1}{v_{slow}} ) \]
Where:
- \(d\) is the thickness of the birefringent material.
- \(\omega\) is the angular frequency
- \(v_{fast}\) is the “fast” velocity resulting from the birefringent material.
- \(v_{slow}\) is the “slow” velocity resulting from the birefringent material.