Polarization equation : lissajous figures

We know there are three types of polarization exist . Those are 

1) . Linearly polarized

2) . Elliptically polarized

3) . Circularly polarized

In any type of polarized light we have specified E-vector direction resulting vector when act on a point charged particles the path in which particle move it is ellipse the light is said to be elliptically polarized .if it is circle or line the the light is called circularly , linearly polarized respectively . 

Proof

Consider 2(why two well it can be more but we considering the source that spontaneously  emitting different amounts of photon per second resulting different amplitude and period of emissions is different resulting different phases for example if time period t=lambda/4c  then phase difference is 90 degree) Electromagnetic wave having same frequency different phases(same wave different amplitude and phase for refraction or reflection emittion etc .) act on a charge particle  . The particle move on the direction of E-vector 

Let X and Y are two different wave emitting from same source having different amplitude and having phase difference different .

from equation 'i'

From equation 'ii'

Putting those values in equation equation  ' iii '

Squaring on both sides and solving we have 


Case i(phase-diff=0 or 180)&(amplitude a=b=A)
Here the light is linearly polarized
Case 2(if phase-dif=90)
In that case we have equation of ellipse hence elliptically polarized .
Case 2.1(phase-dif=90,a=b=a)
This is a equation of circle hence circularly polarized



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