- #1
AAS
- 3
- 0
I want to know the relationship between the optical axis direction of a crystal and the dielectric constants in different directions in an anisotropic material.
How optical axis is related to dielectric tensor?
- Thread starter AAS
- Start date
In summary, The relationship between the optical axis direction of a crystal and the dielectric constants in different directions in an anisotropic material can be derived from the curl relations for E and H, assuming propagating fields are of plane waves. By combining these relations with other equations, the expression ##\mathbf{k} (\mathbf{k}\cdot\mathbf{E})-k^2\mathbf{E} = -\frac{\omega^2}{c^2}\epsilon_r\mathbf{E}## can be obtained, which is used to explain the first expression on page 8 of the provided handout. It should be noted that the presence of ##\mu## on the left side of the equation may
- #2
DrDu
Science Advisor
- 6,357
- 974
See here, especially pages 3 and 8:
http://home.strw.leidenuniv.nl/~keller/Teaching/China_2008/CUK_L03_handout.pdf
http://home.strw.leidenuniv.nl/~keller/Teaching/China_2008/CUK_L03_handout.pdf
- #3
AAS
- 3
- 0
Thank youDrDu said:
- #4
AAS
- 3
- 0
AAS said:
- #5
blue_leaf77
Science Advisor
Homework Helper
- 2,637
- 786
From the curl relations for E and H and the assumption that the propagating fields are of plane waves, one can obtain
$$
\begin{aligned}
\mathbf{k}\times \mathbf{E} = \omega \mu \mathbf{H} \\
\mathbf{k}\times \mathbf{H} = -\omega \mathbf{D}
\end{aligned}
$$
Combining these will yield ##\mathbf{k} (\mathbf{k}\cdot\mathbf{E})-k^2\mathbf{E} = -\omega^2 \mu \mathbf{D}##. Then use the relations like ##\mu = 1/(c^2\epsilon_0)##, ##\mathbf{D} = \epsilon \mathbf{E}##, and ##\epsilon = \epsilon_r\epsilon_0## to transform the RHS into ##-\frac{\omega^2}{c^2}\epsilon_r\mathbf{E}##. So now,
$$
\mathbf{k} (\mathbf{k}\cdot\mathbf{E})-k^2\mathbf{E} = -\frac{\omega^2}{c^2}\epsilon_r\mathbf{E}
$$
Then substitute ##\mathbf{k} = \frac{\omega}{c}n\mathbf{s}## to eliminate ##\omega## in both sides. By taking element-by-element comparison between right and left sides you should see this expression leads to what is written in that slide.
NOTE: I don't think there should be ##\mu## in the LHS of the equation in the slide as its presence will violate the requirement that the LHS and RHS should have the same dimensions.
$$
\begin{aligned}
\mathbf{k}\times \mathbf{E} = \omega \mu \mathbf{H} \\
\mathbf{k}\times \mathbf{H} = -\omega \mathbf{D}
\end{aligned}
$$
Combining these will yield ##\mathbf{k} (\mathbf{k}\cdot\mathbf{E})-k^2\mathbf{E} = -\omega^2 \mu \mathbf{D}##. Then use the relations like ##\mu = 1/(c^2\epsilon_0)##, ##\mathbf{D} = \epsilon \mathbf{E}##, and ##\epsilon = \epsilon_r\epsilon_0## to transform the RHS into ##-\frac{\omega^2}{c^2}\epsilon_r\mathbf{E}##. So now,
$$
\mathbf{k} (\mathbf{k}\cdot\mathbf{E})-k^2\mathbf{E} = -\frac{\omega^2}{c^2}\epsilon_r\mathbf{E}
$$
Then substitute ##\mathbf{k} = \frac{\omega}{c}n\mathbf{s}## to eliminate ##\omega## in both sides. By taking element-by-element comparison between right and left sides you should see this expression leads to what is written in that slide.
NOTE: I don't think there should be ##\mu## in the LHS of the equation in the slide as its presence will violate the requirement that the LHS and RHS should have the same dimensions.
Related to How optical axis is related to dielectric tensor?
1. What is the optical axis and how is it related to the dielectric tensor?
The optical axis is the direction in which light propagates through a material. It is related to the dielectric tensor, which describes the optical properties of a material, by determining the orientation of the principal axes of the tensor.
2. How does the orientation of the optical axis affect the dielectric tensor?
The orientation of the optical axis can affect the dielectric tensor in two ways: it can determine the orientation of the principal axes of the tensor, and it can also determine the anisotropy of the tensor, which describes the difference in properties along different directions.
3. Can the optical axis change in different materials?
Yes, the optical axis can change depending on the material. For example, crystals have a well-defined optical axis due to their ordered atomic structure, while amorphous materials like glass do not have a defined optical axis.
4. How is the dielectric tensor calculated from the optical axis?
The dielectric tensor can be calculated from the optical axis by using the Jones matrix, which relates the input and output electric fields of a material. The orientation of the optical axis and the anisotropy of the tensor are used to determine the elements of the dielectric tensor.
5. What are some real-world applications of understanding the relationship between the optical axis and dielectric tensor?
Understanding the relationship between the optical axis and dielectric tensor is crucial in many fields, such as optics, materials science, and engineering. It is used in the design of optical devices, such as polarizing filters and liquid crystal displays, and in the study of the properties of various materials, such as semiconductors and biological tissues.
Similar threads
-
Electromagnetism
-
Electromagnetism
-
Electromagnetism
-
Electromagnetism
-
Electromagnetism
-
Electrical Engineering
-
Electromagnetism
-
Electromagnetism
-
Electromagnetism