# Direction of extraordinary and ordinary waves (birefringence).

• Silversonic
In summary, the question is about which components of the electric field (E_x or E_z) become the ordinary and extraordinary waves when a linearly polarized wave travels through a plate of uniaxial material with the optical axis in the z-direction. The equations provided show that the ordinary wave has no z-component and propagates in the direction of the incident wave, while the extraordinary wave has a perpendicular electric field and propagates in the x-direction. However, the specific values of k_x, k_y, and k_z cannot be determined without more information.
Silversonic

## Homework Statement

A linearly polarized wave traveling in the y-direction falls on to a plate of uniaxial material of thickness d, and the optical axis is in the z-direction. Which ones of the components $E_x$ or $E_z$ become the ordinary and extraordinary waves inside the plate?

## Homework Equations

Ordinary wave information

$k \circ E = 0$

$\frac{E_x}{E_y} = -\frac{k_y}{k_x}$

$E_z = 0$

Extraordinary wave information$\frac{E_x}{E_y} = \frac{k_x}{k_y}$

$E_z = - (\frac {n_1}{n_3})^2 \frac{k_x^2 + k_y^2}{k_x k_z} E_x$

$E_z = - (\frac {n_1}{n_3})^2 \frac{k_x^2 + k_y^2}{k_y k_z} E_y$

$n_1 , n_3$ are the indices of refraction, the latter of which is for the optical axis.

$k_x, k_y, k_z$ are the wavenumbers in the respective directions for the ordinary/extraordinary waves.

## The Attempt at a Solution

I've looked it up online, and found that when the incident wave is perpendicular to the ordinary axis (the z-axis), the ordinary and extraordinary wave both propagate in the original incident wave's direction, but with different speeds. The problem I'm having is, with the information in the equation section, how does the maths show this is the case (the directions, not the speeds)?

The ordinary wave has no z-component. And, since k.E = 0 we have that the electric field is perpendicular to the ordinary axis and the direction of propagation. But how does this suggests the ordinary wave propagates in the direction of the incident wave i.e. the y-direction? If that is the case, it means $E_x$ becomes the ordinary wave.

The extraordinary wave, I'm not sure how I can pull anything out with the information provided.

There appears to be no way to know $k_x, k_y, k_z$ for the ordinary or extraordinary waves - and thus I can't determine anything about the electric fields. What am I missing?

Last edited:
The only thing I can think of is that the extraordinary wave's electric field is perpendicular to the incident wave's direction, so this means the extraordinary wave must propagate in the x-direction, and thus E_z becomes the extraordinary wave. But I'm not sure how to mathematically back this up.Any help would be greatly appreciated!

## What is birefringence?

Birefringence is the phenomenon in which a material has two different refractive indices for light propagating in two different directions. This results in the splitting of a single light beam into two beams with different polarizations.

## What causes birefringence?

Birefringence is caused by the anisotropic nature of certain materials, meaning that their physical properties are direction dependent. This can be due to their crystal structure, stress, or other factors.

## What is the difference between extraordinary and ordinary waves in birefringence?

In birefringent materials, light can propagate as two distinct waves: the extraordinary (or e-) wave and the ordinary (or o-) wave. The extraordinary wave has a different refractive index than the ordinary wave, resulting in different velocities and directions of propagation.

## How is the direction of extraordinary and ordinary waves determined?

The direction of extraordinary and ordinary waves is determined by the orientation of the birefringent material and the polarization of the incident light. The waves will follow different paths and have different velocities, resulting in a phase difference between them.

## What are some practical applications of birefringence?

Birefringence has many practical applications, including in optical devices such as polarizers, waveplates, and optical filters. It is also used in materials science to study the properties of anisotropic materials and in geology to identify and classify minerals based on their birefringence properties.

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