Interferance term, sum of 2 waves

[fluidistic]

Homework Statement

Consider the following waves: $$\vec E _1 (\vec r , t) =\vec E_1 (\vec r) e^{-i \omega t}$$ and $$\vec E _2 (\vec r , t) =\vec E_2 (\vec r) e^{-i \omega t}$$ where the form of the wavefront isn't specified and where $$\vec E_1$$ and $$\vec E_2$$ are complex vectors which depend on spatial coordinates and the angle of the initial phase. Show that the term of interferance is given by $$I_{12}=\frac{1}{2} (\vec E _1 \cdot \vec E _2 ^* +\vec E _2 \cdot \vec E _1 ^*)$$

Homework Equations

Not sure.

The Attempt at a Solution

Is it just me or the given E fields do NOT depend on the angle of initial phase?!
I took their E fields function, summed them up. It gave me $$e^{-i \omega t} [ \vec E _1 (\vec r ) + \vec E _2 (\vec r )]$$.
Now if I remember well, the intensity of the resultant wave is proportional to the E field squared.
So I squared the expression I just wrote and I reached $$I_{12} = \alpha e^{-2i \omega t} [\vec E _1 ^2 (\vec r ) +\vec E _2 ^2 (\vec r ) +2 \vec E _1 (\vec r) \vec E _2 (\vec r ) ]$$. Now I guess the interference term is $$\alpha \vec E _1 (\vec r) \vec E _2$$ but it does not match the answer.
I realize that the given interference term is worth the sum of the product of the real parts and complex parts of $$\vec E _1$$ and $$\vec E _2$$ and precisely, this is not what happens in my answer.
Where did I go wrong?

 

[vela]

The intensity is given by $I = \mathbf{E}^*\cdot\mathbf{E}$. It's not simply $I=\mathbf{E}^2$.

 

[fluidistic]

The intensity is given by $I = \mathbf{E}^*\cdot\mathbf{E}$. It's not simply $I=\mathbf{E}^2$.

Ok thank you very much. Strangely I can't find it on the Internet. Wikipedia says that it's proportional to $$|E|^2$$. See http://en.wikipedia.org/wiki/Intensity_(physics)#Mathematical_description.
Hmm I'm somehow confused.

 

[vela]

That's right. E is complex, so |E|²≠E².

 

[fluidistic]

That's right. E is complex, so |E|²≠E².

Oh I see, thanks for the clarification!

 

[fluidistic]

That's right. E is complex, so |E|²≠E².

Sorry for bringing this back but I'm still missing something.
If I start from $$e^{-i \omega t} [ \vec E _1 (\vec r ) + \vec E _2 (\vec r )]$$. I can think of it as a complex number of the form $$re^{i \theta}$$, where $$r=\vec E _1 (\vec r ) + \vec E _2 (\vec r )$$ and $$\theta =-\omega t$$.
Then the modulus of E is r. And the modulus squared is $$r^2$$.
Now I'll get something of the form $$I_{12} = \alpha [\vec E _1 ^2 (\vec r ) +\vec E _2 ^2 (\vec r ) +2 \vec E _1 (\vec r) \vec E _2 (\vec r ) ]$$ and I still have no trace of a complex conjugate... Hmm I'll try to continue in this way. If you have any comment, feel free to share knowledge.

 

[vela]

From your first post

where $$\vec E_1$$ and $$\vec E_2$$ are complex vectors

 

[fluidistic]

From your first post

Ahhh, I misunderstood the question, sorry. I thought they meant $$\vec E _1 (\vec r , t)$$ and hence my question regarding the dependence on the angle of initial phase. The dependence was hidden inside $$\vec E _1$$!
Ok I'll rethink the whole problem now.
Thanks once again for pointing that out.

 

[fluidistic]

I almost have it I think.
I reach, starting from and assuming that $$I=E E^*$$ that $$I=|\vec E _1 (\vec r ) |+|\vec E _2 (\vec r )|+\vec E _1 \cdot \vec E _2 ^* +\vec E _2 \cdot \vec E _1 ^*$$.

I realize that the term of interference is $$\vec E _1 \cdot \vec E _2 ^* +\vec E _2 \cdot \vec E _1 ^*$$, but is it well "demonstrated"?
I can argue that if one doesn't know about interference, he will just guess that the intensity at any point in space is the sum of the intensities of the 2 wave sources, namely $$|\vec E _1 (\vec r ) |+|\vec E _2 (\vec r )|$$. While if he does the experience he will see the interference effect and that it can be mathematically described by the term $$\vec E _1 \cdot \vec E _2 ^* +\vec E _2 \cdot \vec E _1 ^*$$.
I wonder if I've solved well the problem. What do you say? I have not used the fact that $$\vec E _i$$, $$i=1,2$$ is dependent on the initial phase.