Electromagnetic Plane Waves

[Rubiss]

Homework Statement

I'm currently trying to understand linear and circular polarization of electromagnetic plane waves. Let's say I have an electric field given by

$$\vec{E}=Acos(kx-\omega t)\hat{x}+Bcos(kx-\omega t - \gamma)\hat{y}$$

A is given and nonzero. I want to find what values of B and gamma that can make the wave linear or circularly polarized.

Homework Equations

I can also write the electric field as

$$\vec{E}=Re \big((A\hat{x}+Be^{-i\gamma}\hat{y})e^{i(kx-\omega t)}\big)$$

The Attempt at a Solution

For the linear polarization, I'm thinking I can either make B=0, or gamma equal to n*pi if B is not equal to zero. Can anyone comment on my thinking?

For circular polarization, I'm thinking I need to have A=B and gamma equal to n*pi/2 for odd n. I will have right handed circular polarization if n=3,7,11,... and have left handed circular polarization if n=1,5,9,... Is this thinking correct?

Is there an easier way to do this that I am not seeing?

 

[rude man]

Are you sure it's not cos(kz - wt) etc.? Because if not this is not a possible e-m wave.

 

[rude man]

Assuming propagation in the z direction, you can solve this by writing the E field at z = 0, writing the x and y components of E and eliminating t between them. The result is y(x,phi,A,B) which then makes it obvious the restrictions on B and phi you must impose to effect linear or circular polarization. You could even go further and do the same for elliptical polarization for extra credit ...

 

[rude man]

Homework Statement

I'm currently trying to understand linear and circular polarization of electromagnetic plane waves. Let's say I have an electric field given by

$$\vec{E}=Acos(kx-\omega t)\hat{x}+Bcos(kx-\omega t - \gamma)\hat{y}$$

A is given and nonzero. I want to find what values of B and gamma that can make the wave linear or circularly polarized.

Homework Equations

I can also write the electric field as

$$\vec{E}=Re \big((A\hat{x}+Be^{-i\gamma}\hat{y})e^{i(kx-\omega t)}\big)$$

OK. I never did respond to you directly. Here goes:

The Attempt at a Solution

For the linear polarization, I'm thinking I can either make B=0, or gamma equal to n*pi if B is not equal to zero. Can anyone comment on my thinking?

Right.

For circular polarization, I'm thinking I need to have A=B and gamma equal to n*pi/2 for odd n. I will have right handed circular polarization if n=3,7,11,... and have left handed circular polarization if n=1,5,9,... Is this thinking correct?

Almost right. B can be + or -A. The choice of sign determines right or left circular polarization. That makes it tantamount to choosing n for γ the way you did.

Is there an easier way to do this that I am not seeing?

Hard to say since you didn't tell us how you got your results in the first place ...

 

[paranormal]

Your thinking is correct for both linear and circular polarization. For linear polarization, B=0 will result in a linearly polarized wave along the x-axis, while setting gamma equal to n*pi will rotate the polarization by n*pi/2, resulting in a linearly polarized wave along the y-axis.

For circular polarization, setting A=B and gamma=n*pi/2 will result in a circularly polarized wave with the same amplitude in both the x and y directions, but with a phase difference of n*pi/2 between them. This will result in right-handed circular polarization for odd values of n and left-handed circular polarization for even values of n.

There is no easier way to determine the values of B and gamma for linear and circular polarization, as it is dependent on the desired polarization and the given electric field. However, you can always use the general equation for an electric field in terms of linear and circular polarization to verify your results. Good job on your thinking and analysis!