Angle of refraction, Refractive Indices, and D Vectors

[Joystar77]

Homework Statement

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.

Homework Equations

OPL difference = triangle d n o

Phase difference = triangle does not equal 0 = k o triangle = 2 triangle/ x d (n o - n e)

The Attempt at a Solution

Sin Oi = n o sin 0 n e

O i = 45 degrees

Sin Oi = 1/ Square Root 2

Sin O no = 1 / 1.54264 Square Root 2

Sin O ne = 1 / 1.55171 Square Root 2

Triangle 0 = Sin + 1/ 1.54264 Square Root 2 - Sin + 1/ 1.55171 Square Root 2

Are any of these right for a, b, or c? If not, can somebody please explain step-by-step how to do these?

 

[anathema]

Thank you for your question. To solve these problems, we will need to use Snell's law, which states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the refractive indices of the two media. We will also need to use the equations given in the problem, including the Sellmeier equation and the equations for OPL and phase difference.

a.) In this case, since the optic axis is parallel to the plane of incidence and the surface of the crystal, the light will travel through the crystal along the optic axis. This means that the D vectors for both the o- and e- waves will be parallel to the optic axis. To find the angle of refraction, we can use Snell's law:

Sin Oi = n o sin Or

Solving for the angle of refraction, we get:

Or = sin^-1 (Sin Oi / n o) = sin^-1 (1 / 1.54264 Square Root 2)

Using a calculator, we find that Or is approximately 41.3 degrees. To find the refractive indices, we can use the Sellmeier equation:

n o = Square Root (1 + B1 L^2 / (L^2 - C1) + B2 L^2 / (L^2 - C2) + B3 L^2 / (L^2 - C3))

n e = Square Root (1 + B1 L^2 / (L^2 - C1) + B2 L^2 / (L^2 - C2) + B3 L^2 / (L^2 - C3))

Plugging in the given values for L and the coefficients B1, B2, B3, C1, C2, and C3, we can solve for n o and n e. I will leave the calculation to you.

b.) In this case, since the optic axis is perpendicular to the surface of the crystal, the light will travel through the crystal perpendicular to the surface. This means that the D vectors for both the o- and e- waves will be perpendicular to the optic axis. To find the angle of refraction, we can again use Snell's law:

Sin Oi = n o sin Or

Solving for the angle of refraction, we get:

Or = sin^-1 (Sin Oi / n