Concept inventory for Birefringence

Concept inventory for Birefringence

1. The k-vector is perpendicular to the wavefronts. However, the wave direction may not be parallel to the k-vector direction
2. In all cases, the true 'direction' of the wave is described by the Poynting vector
3. When the beam is normally incident with double refraction, the k-vector (of either polarization component) DOES NOT REFRACT AT ALL
4. When the beam is normally incident with double refraction, the Poynting vector of the e-wave propagates at an angle w.r.t. the surface normal, hence power propagates at an angle
5. The Poynting vector is normal (perpendicular) to the electric field vector. After all it is defined as a cross-product of the electric field vector and the magnetic field.
6. The k-vector is normal to the displacement field vector $\overset{\rightharpoonup }{D}\equiv \overleftrightarrow{\epsilon }\overset{\rightharpoonup }{E}$
7. Snell's Law still applies -- Snell's law is simply a statement regarding the conservation of the k-vector. It says that the portion of the k-vector parallel to the surface is conserved
8. The walk-off (angle) associated with double refraction is due to the power flow in the direction of the Poynting vector and is also the angle between $\overset{\rightharpoonup }{E}$ and $\overset{\rightharpoonup }{D}$, which are NOT parallel for the e-wave.$\overset{\rightharpoonup }{E}$ and $\overset{\rightharpoonup }{D}$ are parallel for the o-wave. The walk-off only happens in the crystal -- it stops happening as the wave exits the crystal. The k-vector was always normal to the surface (if it started that way) and remains that way at the output. Outside the crystal, the Poynting vector becomes parallel to the k-vector again.

Here is a picture explaining what you will see in the following animations. The left hand of both images represents vacuum. The right hand of both images represents an anisotropic medium. Specifically, the medium is a birefringent, uniaxial crystal with optic axis indicated by the dashed line. Remember that in all directions perpendicular to the optic axis the index of refraction is n_o. In the direction of the optic axis, the index of refraction is n_e.

A plane wave propagates from left to right and illuminates a single emitter (dipole) located on boundary between the two media. The plane wave is polarized in the vertical direction. Because the medium is birefringent, we expect that the phase velocity of the light emitted by the dipole to depend on propagation direction.

From previous animations we should expect the dipole to emit light spherically. However, because the wave speed is different parallel to the optic axis than it is perpendicular to the optic axis, the waves will likely NOT be spherical. It turns out the waves are elliptical.

I've drawn a black ellipse in the animation to highlight the shape of the wave in the material. So the next step is to add a few more emitters to model the interface more accurately.

Things are getting a bit more complicated. It turns out that we can make some sense of this by adding more dipoles

This animation shows that all of the elliptical waves add coherently in the FORWARD direction. That is -- the wavefronts are in the same direction as the input! We have already established that the k-vector is perpendicular to the wavefronts ($\overset{\rightharpoonup }{k}\equiv \nabla \varphi$) so we can take this to show that the wave does not refract in the usual sense.

As we have discussed previously, the k-vector does not refract. But the Poynting vector does change direction upon entering the material. Let's take a look at the Intensity by time averaging the Poynting vector. We will use

$I\equiv \left\langle \left| \overset{\rightharpoonup }{S} \right| \right\rangle _T=\langle S\rangle _T=\epsilon _0c\left\langle E^2\right\rangle {}_T$

to calculate the intensity and we'll plot that in the birefringent media on the right.

Now, if you look carefully you can see a slight, upward shift of the wave in the material but things are a little unclear. To simplify this, we will go back to only a few emitters to 'simulate' a narrow beam of light incident on the material.

And there we have it! The light emitted backwards into vacuum goes more-or-less straight backwards (it spreads out because there are only a few emitters). The intensity of the light emitted into the birefringent material is clearly propagating at an angle. In fact, the angle the intensity makes with horizontal axis is in fact, the walkoff angle!