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.
a) In this case, the optic axis is parallel to the plane of incidence and parallel to the surface of the crystal.
To find the angle of refraction, we can use Snell's law, which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the inverse ratio of the refractive indices:
n_o * sin(θ_i) = n * sin(θ_r)
Given that the angle of incidence (θ_i) is 45 degrees, and the refractive indices (n_o and n_e) are given in the question, we can substitute these values into the equation to find the angle of refraction (θ_r):
1.54264 * sin(45) = 1.55171 * sin(θ_r)
θ_r can be found by solving this equation. In this case, the angle of refraction is approximately 44.874 degrees.
The direction of the D vectors for the o- and e-waves inside the crystal can be described as follows:
- The D vector for the o-wave will be parallel to the optic axis, which in this case is parallel to the surface of the crystal.
- The D vector for the e-wave will be perpendicular to the optic axis and parallel to the surface of the crystal.
b) In this case, the optic axis is perpendicular to the surface of the crystal.
Using the same approach as in case (a), we can find the angle of refraction. Again, the angle of incidence (θ_i) is 45 degrees, and the refractive indices (n_o and n_e) are given in the question. Solving the same equation as before:
1.54264 * sin(45) = 1.55171 * sin(θ_r)
θ_r can be found by solving this equation. In this case, the angle of refraction is approximately 44.826 degrees.
The direction of the D vectors for the o- and e-waves inside the crystal can be described as follows:
- The D vector for the o-wave will be perpendicular to the optic axis and parallel to the surface of the crystal.
- The D vector for the e-wave will be parallel to the optic axis, which in this case is perpendicular to the surface of the crystal.
c) In this case, the optic axis is perpendicular to the plane of incidence.
Again, we can use the same approach as in the previous cases to find the angle of refraction. In this case, the angle of incidence (θ_i) is 45 degrees, and the refractive indices (n_o and n_e) are given in the question. Solving the same equation as before:
1.54264 * sin(45) = 1.55171 * sin(θ_r)
θ_r can be found by solving this equation. In this case, the angle of refraction is approximately 44.876 degrees.
The direction of the D vectors for the o- and e-waves inside the crystal can be described as follows:
- The D vector for the o-wave will be parallel to the plane of incidence and perpendicular to the optic axis.
- The D vector for the e-wave will be parallel to the plane of incidence and parallel to the optic axis.
To solve these problems, we need to use Snell's law, which relates the angles of incidence and refraction to the refractive indices of the two media. Snell's law states:
n1 * sin(theta1) = n2 * sin(theta2)
Where n1 and n2 are the refractive indices of the two media, theta1 is the angle of incidence, and theta2 is the angle of refraction.
a.) In this case, the optic axis is parallel to the plane of incidence and also parallel to the surface of the crystal. Since the optic axis is parallel to the plane of incidence, the ordinary and extraordinary rays will travel in the same direction.
The angle of incidence (theta1) is given as 45 degrees. The refractive index for the ordinary wave, n_o, is 1.54264, and for the extraordinary wave, n_e, is 1.55171.
Using Snell's law, we can calculate the angle of refraction (theta2) for both the ordinary and extraordinary waves. Since the ordinary and extraordinary rays travel in the same direction, the angle of refraction will be the same for both waves.
n_o * sin(theta1) = n_o * sin(theta2)
1.54264 * sin(45) = 1.54264 * sin(theta2)
Solving for theta2, we find that the angle of refraction is also 45 degrees.
The D vector for the ordinary wave will be parallel to the direction of propagation, while the D vector for the extraordinary wave will also be parallel to the direction of propagation since the optic axis is parallel to the plane of incidence.
b.) In this case, the optic axis is perpendicular to the surface of the crystal. This means that the ordinary ray will have a different direction of propagation compared to the extraordinary ray.
The angle of incidence (theta1) is given as 45 degrees. The refractive index for the ordinary wave, n_o, is 1.54264, and for the extraordinary wave, n_e, is 1.55171.
Using Snell's law, we can calculate the angle of refraction (theta2) for both the ordinary and extraordinary waves. Since the optic axis is perpendicular to the surface of the crystal, the ordinary and extraordinary rays will have different angles of refraction.
n_o * sin(theta1) = n_o * sin(theta2)
1.54264 * sin(45) = 1.54264 * sin(theta2_o)
1.55171 * sin(45) = 1.55171 * sin(theta2_e)
Solving for theta2_o, we find that the angle of refraction for the ordinary wave is approximately 40.647 degrees, and for theta2_e, the angle of refraction for the extraordinary wave is approximately 40.532 degrees.
The D vector for the ordinary wave will be perpendicular to the direction of propagation, while the D vector for the extraordinary wave will be parallel to the direction of propagation since the optic axis is perpendicular to the surface of the crystal.
c.) In this case, the optic axis is perpendicular to the plane of incidence. This means that the ordinary ray will have a different direction of propagation compared to the extraordinary ray.
The angle of incidence (theta1) is given as 45 degrees. The refractive index for the ordinary wave, n_o, is 1.54264, and for the extraordinary wave, n_e, is 1.55171.
Using Snell's law, we can calculate the angle of refraction (theta2) for both the ordinary and extraordinary waves. Since the optic axis is perpendicular to the plane of incidence, the ordinary and extraordinary rays will have different angles of refraction.
n_o * sin(theta1) = n_o * sin(theta2_o)
1.54264 * sin(45) = 1.54264 * sin(theta2_o)
1.55171 * sin(45) = 1.55171 * sin(theta2_e)
Solving for theta2_o, we find that the angle of refraction for the ordinary wave is approximately 40.647 degrees, and for theta2_e, the angle of refraction for the extraordinary wave is approximately 40.532 degrees.
The D vector for the ordinary wave will be parallel to the direction of propagation, while the D vector for the extraordinary wave will be perpendicular to the direction of propagation since the optic axis is perpendicular to the plane of incidence.