Expressing plane wave as superposition

[Ishida52134]

Homework Statement

So, given a unpolarized monochromatic plane wave E = summation ai cos(kz - wt + bi), i from 1 to N where b is a phase constant. how would you describe this as the superposition of a right handed and left handed polarized beam?

Homework Equations

Er = Acos(kz-wt+phi1) + A sin(kz-wt+phi1)
El = Acos(kz-wt+phi2) - Asin(kz-wt+phi2)

The Attempt at a Solution

I know if I add both up I would get a linearly polarized wave.
I'm not sure how to write the sum as such a summation. Would you use the fact that any finite linear combinations of cosine terms can add up to the sum which would show that the summation can be written in terms of one cosine function?

 

[Simon Bridge]

Have you tried expanding each term in the sum as a linear sum of specific Er and El.

 

[Ishida52134]

I think it'd be the superposition of one right and one left circular polarized beam that describes the whole summation.

 

[Simon Bridge]

Good luck with that.

 

[paranormal]

I would approach this problem by first identifying the key components of the given plane wave expression. We have a summation of ai cos(kz - wt + bi), which can be rewritten as ai cos(kz - wt) cos(bi) + ai sin(kz - wt) sin(bi). This can be further simplified as Acos(kz - wt + phi) where phi = bi is the phase constant.

Now, to express this as the superposition of a right handed and left handed polarized beam, we can use the equations given in the homework section. We know that a right-handed polarized beam is given by Er = Acos(kz-wt+phi1) + A sin(kz-wt+phi1) and a left-handed polarized beam is given by El = Acos(kz-wt+phi2) - Asin(kz-wt+phi2).

By comparing these equations with our simplified form of the given plane wave expression, we can see that phi1 = bi and phi2 = -bi. Therefore, we can express the plane wave as a superposition of a right-handed and left-handed polarized beam as:

E = Er + El = Acos(kz-wt+bi) + A sin(kz-wt+bi) + Acos(kz-wt-bi) - Asin(kz-wt-bi)

This can also be written as a summation of two terms, one for the right-handed polarized beam and one for the left-handed polarized beam:

E = summation ai cos(kz - wt + bi) + ai cos(kz - wt - bi)

Thus, we have successfully expressed the given plane wave as the superposition of a right-handed and left-handed polarized beam.