Finding an Orthornomal Polarization?

[Noone1982]

Say I have a polarization [3i, 2], How do I find a polarization that is orthogonal?

I know that,

$$AA^{\cdot }\; +\; BB^{\cdot }\; =\; 0$$

But my problem is that it yields one equation and two unknowns which I can't solve for. Furthermore, I am a bit confused on the representation of [3i,2] I understand [1,(+/-)i] is circularly polarized light but what about when the i is atop?

 

[Mindscrape]

So, you know that something is orthogonal when its inner product is zero. Find all A' and B' such that this the condition is satisfied. Even though you have one equation and two unknowns, that just indicates that there are infinite solutions depending on a free variable.

You are confused on the mathematical representation or the physical representation?

 

[m2003]

To find an orthogonal polarization, you can use the Gram-Schmidt process. This involves finding a vector that is orthogonal to your given polarization [3i, 2] and then normalizing it to make it an orthonormal vector.

To find this orthogonal vector, you can start by choosing any arbitrary vector that is not parallel to [3i, 2]. This can be [2, -3i] for example. Then, you can use the formula

v' = v - (v · u)u

where v is your given polarization and u is the arbitrary vector you chose. This formula will give you an orthogonal vector v' to your given polarization.

In this case, your orthogonal polarization would be [2, -3i]. To normalize it, you can divide each component by its magnitude, which would be sqrt(13) in this case. So your final orthonormal polarization would be [2/sqrt(13), -3i/sqrt(13)].

As for the representation of [3i, 2], it is a vector with a complex component. This represents a linearly polarized light with a phase difference of pi/2 between the electric and magnetic field components. The i on top indicates the direction of rotation of the electric field.