Question: What does it mean when we say that the monochromatic wave is un-polarized? Is this kind of wave actually possible? What would the equation of such a wave look like? Since we can consider also that there are no ideal monochromatic waves. How are these un-polarized waves generated?

> 
> Ex=│()│cos(− + (t))
> 
> 
> Ey= │()│cos(− + (t))
> 
> 
> Δ(t)=(t)−(t)
> 
> 
> 

(t)=time varying phase offset of x polarized wave and

(t)=time varying phase offset of y polarized wave

When these waves are superposed do we get any specific polarization?

Is it Possible to have a phase offset varying with respect to space?

> 
> Δ(z)=(z)−(z)
> 
> 
>

Answer: 
* Monochromatic means that the light possesses a single wavenumber $k = 2\pi/\lambda$.
* The polarisation vector $\vec \epsilon$ describes in which direction the electric field $\vec E(z,t)$ oscillates. Unpolarised means that the polarisation changes **randomly**.

Hence, an unpolarised, monochromatic wave propagating in the $z$ direction is a wave described by
$$
\vec E(z,t) = E\_0 \cos(\omega t - kz) \cdot\vec \epsilon(t)
$$
where the polarisation vector changes it's direction randomly within the $(x,y)$-plane.

Physically the polarisation vector is unable to change its direction randomly within an *arbitrary small* amount of time. Hence, if we consider a "small enough" time interval $\Delta t$ the polarisation will always have a preferred direction, and therefore a polarisation. However, because

1. visible light has an oscillation frequency of the order of $\nu \approx 5 \cdot 10^{14}Hz$, and
2. the measurement time $\Delta t$ is for most applications larger than $1\mu s$,

almost all measurements are done by averaging millions of oscillations. Therefore, if we consider a thermal light source (containing "many" point-like sources) the averaged polarisation is to a good approximation random within the measurement time interval $\Delta t$. Therefore, such a source is often considered to possess a random polarisation.
Light from a single atom is polarized.
The electric field for unpolarized light is given by a sum of the radiation
from many uncorrelated individual atoms, so no particular polarization direction
can be defined.