1. Polarization of EM waves
The polarization of a uniform plane wave refers to the time-varying behaviour of the electric field vector at fixed point in space.
Consider, for example, a uniform plane wave travelling in the z-direction. With the E and H vectors lying in the x-y plane.
Case 1:
If Ey=0, and only Ex is present:
The wave is said to be polarized in the x direction.
If Ex=0, and only Ey is present:
The wave is said to be polarized in the y direction.

2. If both Ex and Ey are present, and are in phase (=0, 2, 4
If both Ex and Ey are present, and are in phase (=0, 2, 4....) the resultant electric vector has a direction, dependent on the relative magnitude of Ex and Ey.
The resultant vector will make the angle tan-1(Ey/Ex).
In all the above cases the wave is said to be linearly polarized because the resultant vector is constant with time.
Case 2:
If both Ex and Ey are not in phase (=(2n+1)/2, n=0, 1, ), i.e. their maximum values (of amplitude) reach at different time, the direction of resultant electric vector will vary.

3. The locus of the end point of the resultant E will be an ellipse and the wave is said to be elliptically polarized.
Case: 3
When Ex and Ey have equal magnitudes and not in phase i.e.(=(2n+1)/2, n=0, 1, ), the locus of the resultant E will be an circle and the wave is said to be circularly polarized.

4. Equation (1) can be written as:
Let the two plane polarized EM waves may superimpose. One has Electric vector, Ex in X direction and other has electric vector, Ey in Y direction, and both are propagating in Z direction.
Where a and b are the amplitude of the waves and  is the phase difference between them.
Equation (1) can be written as:

5. From 2nd equation:
From above equation we can find:
Put these values in equation 3: or or

6. or or
This equation in general represents an ellipse. Therefore in this case the resultant will be elliptically polarised light. But actual nature of resultant wave can be given by following cases:
Case I:
n: integer
then

7. So equation 4 will be: or or or
Which is the equation of a straight line. Thus emergent ray in this case will be plane polarised light.

8. Case II:
then
So equation 4 will be:
Which is the equation of an ellipse. Thus emergent ray in this case will be elliptically polarised light.

9. Case III: with a=b
So equation 4 will be:
Which is the equation of a circle of radius a. Thus emergent ray in this case will be circularly polarised light.