What Is the Angle of the D Vector in a Uniaxial Crystal?

[v_pino]

Homework Statement

Here you will calculate the direction of energy flow in a uniaxial crystal. The crystal has dielectric constant $$\varepsilon$$ in the x and y directions and $$\varepsilon_z$$ in the z direction.

(a) Assume the wavevector makes an angle $$\theta$$ relative the z-axis of the crystal. What is the angle of the D vector with respect to this axis? Now, use the inverse of the dielectric tensor to obtain the direction of E from the direction of D.

Homework Equations

The Attempt at a Solution

$$\varepsilon = \begin{bmatrix} \varepsilon & 0 & 0 \\ 0 & \varepsilon & 0 \\ 0 & 0 & \varepsilon_z \end{bmatrix}$$

$$\begin{bmatrix} D_x\\ D_y\\ D_z\\ \end{bmatrix} = \begin{bmatrix} \varepsilon & 0 & 0 \\ 0 & \varepsilon & 0 \\ 0 & 0 & \varepsilon_z \end{bmatrix} \begin{bmatrix} E_x\\ E_y\\ E_z \end{bmatrix}$$

And I know that birefringe causes the wave to separate such that:

$$D_e$$ is in the y-z plane perpendicular to S, the poynting vector and $$D_0$$ is parallel to x-axis.

Can you give me some hints or reading materials as to solving the problem? Thank you.

 

[mark726]

Hi there,

To solve this problem, we can start by considering the direction of the wavevector in the crystal. Since it is making an angle \theta relative to the z-axis, we can use basic trigonometry to determine the components of the wavevector in the x, y, and z directions. This will give us an idea of the direction the wave is traveling in the crystal.

Next, we can use the inverse of the dielectric tensor to obtain the direction of the electric field (E) from the direction of the displacement field (D). This can be done by solving the equation D = \varepsilon E for E. This will give us the direction of the electric field in terms of the components of the displacement field.

Finally, to determine the direction of energy flow, we can use the Poynting vector, which is given by \vec{S} = \vec{E} \times \vec{H} . This vector represents the direction and magnitude of energy flow in an electromagnetic wave. We can use this to determine the direction of energy flow in the crystal, taking into account the direction of the electric and magnetic fields.

Some recommended reading materials for this topic include textbooks on electromagnetism and optics, as well as online resources such as Khan Academy or HyperPhysics. I hope this helps and good luck with your calculations!