# Electromagnetic Plane Waves

1. Apr 24, 2013

### Rubiss

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

I can also write the electric field as

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

3. 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?

2. Apr 26, 2013

### rude man

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

3. Apr 26, 2013

### 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 ...

4. Apr 26, 2013

### rude man

OK. I never did respond to you directly. Here goes:
Right.
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.
Hard to say since you didn't tell us how you got your results in the first place ...