Interference of circularly polarized waves

[carllacan]

Homework Statement

Two (left) circularly polarized monochromatic plane waves S2 and S3 arrive at the xy plane with an incidence angle θ (see figure). Find the intensity of the light projected in the xy plane.

(Nevermind S1, it is only mentioned in another part of the problem)

Homework Equations

The Attempt at a Solution

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I've written down the expression for a monochromatic plane wave circularly polarized using $$\vec e_1 = sinθ\vec i + cos θ \vec j $$ $$\vec e_2 = \vec j$$ as the basis for the polarization vector for S2 and $$\vec e_1 = -sinθ\vec i + cosθ \vec j $$ $$\vec e_2 = \vec j$$ for S3. Then the electromagnetic vector at the xy plane its just the sum of them both.

Now the intensity would be $$ I = \frac{εc} {n} \langle \vec E_r^2 \rangle = \frac{εc} {c} \vec E · \vec E^* = \frac{εc} {c} (\vec E_1 + \vec E_2)·(\vec E_1^* + \vec E_2 ^*)$$, but to get to this point every source I've found has assumed that both waves are linearly polarized, so I'm not sure if it would be the same with other polarizations. I don't see why not, but I would like to confirm it before going on.

 

[blue_leaf77]

First of all for circular polarization, the TE and TM components of the light are dephased by 90⁰, which doesn't seem to manifest in your expression for the unit vectors. Second, since the incoming beams are not parallel to the z axis, therefore you must expect that the E field has a component along the z-direction.