How Do I Determine the Polarization State and Azimuthal Angle of a Wave?

[roam]

Homework Statement

Determine the state of polarization of the following wave and its azimuthal angle:

$E= \sqrt{3} E_0 \cos(kz-\omega t) \hat{x} + E_0 \cos(kz- \omega t + \pi) \hat{y}$

The Attempt at a Solution

How do I calculate the azimuthal angle?

My textbook only says: "the azimuthal angle is defined as the angle between the plane of vibration and the plane of incidence". Since no diagrams are provided I'm not sure if I understand it correctly... Is this the angle it makes with the x-axis if it is linearly polarized (or if elliptically polarized, the angle of the major axis with respect to the x-axis)?

I think the state of polarization is linearly polarized since E_y lags E_x by π, right? Also the E_x has a greater amplitude than E_y by a factor of √3.

Any help is greatly appreciated.

 

[UltrafastPED]

For definitions see http://en.wikipedia.org/wiki/Plane_of_incidence

 

[roam]

For definitions see http://en.wikipedia.org/wiki/Plane_of_incidence

It doesn't give the definition of the "azimuthal angle". I believe it's the angle θ I've marked in the attached diagram. How can it be calculated?

 

[roam]

How do I calculate azimuthal angle for this wave?

I know it's not 45° because $E_x \neq E_y$. Is it valid to use trig like this:

$\tan \theta = \frac{E_y}{E_x} = \frac{E_0}{\sqrt{3} E_0} \implies \theta = 30°$

 

[paranormal]

I would suggest using the Jones calculus method to determine the state of polarization and azimuthal angle of the given wave. This method involves using a matrix representation of the electric field components to calculate the polarization state and angle.

To calculate the azimuthal angle, you can use the following formula:

θ = tan^-1(Ey/Ex)

Where Ey and Ex are the amplitudes of the electric field components along the y and x axes, respectively.

In this case, the state of polarization is indeed linear, as Ey lags behind Ex by π, indicating a phase difference of 180 degrees. The azimuthal angle can be calculated as:

θ = tan^-1(1/√3) = 30 degrees

This means that the wave is linearly polarized at an angle of 30 degrees with respect to the x-axis.

It is important to note that the azimuthal angle can also be calculated using the complex representation of the wave, known as the Stokes parameters. However, the Jones calculus method is simpler and more straightforward in this case.

I hope this helps in understanding the concept of polarization and azimuthal angle.