At the He-Ne laser wavelength (L= 632.8 nm) the refractive indices of crystal quartz are n o = 1.54264 and n e = 1.55171 calculated from its Sellmeier equation. The laser is incident from the air onto the surface of crystal quartz at an angle of incidence of 45 degrees. For each of the following three cases, please find the angle of refraction, find the refractive indices, and briefly describe the direction of the D vectors for the o- and e- waves inside the crystal.
a.) The optic axis is parallel to the plane of incidence, and is also parallel to the surface of the crystal.
b.) The optic axis is perpendicular to the surface of the crystal.
c.) The optic axis is perpendicular to the plane of incidence.
To find the angle of refraction and the refractive indices in each of the given cases, we will need to apply Snell's law and consider the different behaviors of ordinary (o) and extraordinary (e) waves in crystal quartz.
a.) The optic axis is parallel to the plane of incidence and parallel to the surface of the crystal:
In this case, the ordinary and extraordinary waves will follow different paths due to the birefringent nature of crystal quartz.
To find the angle of refraction, we can use Snell's law: n1 * sin(theta1) = n2 * sin(theta2), where n1 and n2 are the refractive indices of the initial and final mediums, respectively, and theta1 and theta2 are the angles of incidence and refraction, respectively.
Since we are going from air (n1 ≈ 1) to crystal quartz (n2), the equation becomes: sin(theta1) = (n2/n1) * sin(theta2).
Since the angle of incidence is 45 degrees, we have: sin(45) = (n2) * sin(theta2) / 1.
Now, substitute the given refractive index n2 for the corresponding polarized wave (o or e) obtained from the Sellmeier equation: n2 = no (ordinary wave) or n2 = ne (extraordinary wave).
b.) The optic axis is perpendicular to the surface of the crystal:
In this case, the ordinary and extraordinary waves will propagate with the same refractive index, as the optic axis is perpendicular to the surface. Thus, both waves will have the refractive index of the ordinary wave (no).
c.) The optic axis is perpendicular to the plane of incidence:
For this case, it implies that the optic axis is perpendicular to the plane defined by the incident ray and the normal to the surface. The ordinary and extraordinary waves will propagate with different refractive indices.
Again, we can use Snell's law to find the angle of refraction, considering the ordinary (no) and extraordinary (ne) wave refractive indices from the Sellmeier equation.
Regarding the direction of the D vectors, the ordinary wave (o-wave) will have its D vector parallel to the plane of incidence, while the extraordinary wave (e-wave) will have its D vector perpendicular to the plane of incidence. This behavior arises due to the anisotropic nature of crystal quartz, where the optic axis determines the orientation of the D vectors.