Major and minor axes of elliptically polarized light

[Robsta]

Homework Statement

Consider an elliptically polarized beam of light propagating along the z axis for which the E field components at a fixed position z are:

E_x = E₀cos(ωt) and E_y = E₀cos(ωt +φ)

Find the major and minor axes of the ellipse in terms of E₀ and φ and sketch the ellipse in the E_x-E_y plane.

Homework Equations

The Attempt at a Solution

I know that elliptically polarised light is formed by two waves with perpendicular polarisations and a phase shift of 90°. They have unequal amplitudes (if they had equal amplitudes, then it would be circularly polarized).

The major axis of the ellipse will be along the polarization axis of the wave with the bigger amplitude.
The minor axis will be along the polarization axis of the wave with the smaller amplitude.

So perhaps this is a maximisation problem?
Or maybe there's something to do with a cross product that I'm missing.

If we say that the major axis is theta from the x axis, then:

E_x+θ = E_Bcos(ωt - θ)
E_y+θ = E_scos(ωt - θ + φ)

Where E_s stands for the smaller of the two amplitudes and E_B stands for the bigger of the two amplitudes.

In this new frame, the phase difference must be 90 degrees, but doesn't that make φ = 90°? But if that were the case, then the unrotated frame would be circularly polarized. I'm very confused.

 

[Robsta]

Looking at this more, does φ have to be 90 degrees always? Two waves have the same phase difference no matter what frame you view them from, right?

 

[Robsta]

Okay I've drawn a diagram and had another thought. if E_x and E_y describe points on an ellipse with equal radius, they must be at ±π/4 relative to the major and minor axis.

So θ = ±π/4

Here's the diagram I did to convince myself that this must be the case.

Trying to work out what to do next now.