Measuring intensity of superposed waves using complex amplitudes

[McLaren Rulez]

Hi,

Suppose we have the x and y components of the electric field being described as $(E_{x}e^{i(kz-\omega t)}, E_{y}e^{i(kz-\omega t +\phi)})$, what is the intensity?

I think the correct answer is $E_{x}^{2} +E_{y}^{2} + 2E_{x}E_{y}\cos\phi$. However, I am not sure how to deal with this using the Jones formalism. In that, the intensity is given by $E^{\dagger}E$ which would give

$$\begin{pmatrix} E_{x}e^{-i(kz-\omega t)} & E_{y}e^{-i(kz-\omega t +\phi)} \end{pmatrix} \begin{pmatrix} E_{x}e^{i(kz-\omega t)}\\ E_{y}e^{i(kz-\omega t +\phi)}\end{pmatrix} =E_{x}^{2} +E_{y}^{2}$$

Clearly, the above answer is independent of the relative phase and I think it cannot be right because of that. So what is the correct way to calculate the intensity using Jones formalism? Thank you

 

[mfb]

How do you get your first result?
If you flip the sign of E_y, you modify your calculated intensity, which is wrong.

 

[McLaren Rulez]

The first result was using

$I = (E_{x}e^{-i(kz-\omega t)}+ E_{y}e^{-i(kz-\omega t +\phi)})(E_{x}e^{i(kz-\omega t)}+ E_{y}e^{i(kz-\omega t +\phi)})$

Now that I think of it, this doesn't seem correct either. What is the correct expression for intensity?

 

[mfb]

E_x^2 + E_y^2 should be correct.

 

[McLaren Rulez]

Are you sure about that? Is the phase completely irrelevant in the intensity calculation?