What is the Condition for Interference of Two Plane Waves?

[KFC]

Homework Statement

Find out the condition for two-plane-wave interference. The two plane waves are given as

$$\vec{E}_1 = E_0\vec{A}\exp(ikz - i\omega t) + c.c.$$

$$\vec{E}_2 = E_0\vec{B}\exp(ikz - i\omega t + i\phi) + c.c.$$

where $$\vec{A}$$ and $$\vec{B}$$ are complex. All other variables are real.


2. The attempt at a solution
Here is how I do the problem. I first let

$$\vec{A} = A\exp(i\alpha)$$

$$\vec{B} = B\exp(i\gamma)$$

and define
$$F = kz - \omega t, \qquad G = kz - \omega t + \phi$$
so

$$\vec{E}_1 = E_0A \left(\exp[i(F + \alpha)] + \exp[-i(F+\alpha)]\right) = 2E_0 A\cos(F+\alpha)$$

$$\vec{E}_2 = E_0B \left(\exp[i(G + \beta)] + \exp[-i(G+\beta)]\right) = 2E_0 B\cos(G+\beta)$$

Now, both $$\vec{E}_1$$ and $$\vec{E}_2$$ become real. So I can square the total field $$\vec{E} = \vec{E}_1 + \vec{E}_2$$ directly

$$E^2 = E_1^2 + E_2^2 + 2E_1E_2 = 4E_0^2 A^2\cos^2(F+\alpha) + 4E^2_0 B^2\cos^2(G+\beta) + 8E_0^2AB\cos(F+\alpha)\cos(G+\beta)$$

But I remember the interference term should only contain the

$$E_0^2AB\cos(\phi+\alpha-\beta)$$

 

[turin]

What do you mean by "the condition for two-plane-wave interference"? They always interfere, regardless of any specific phases and whatnot.

Are these waves supposed to be going in exactly the same direction? If so, this problem may be quite easy (depending on what "the condition for two-plane-wave interference" means.)