Randomly Polarized Light
Randomly polarized light: The electric field associated with a plane electromagnetic wave can be in two mutually perpendicular plane (maintaining the direction of electric field perpendicular to the direction of propagation). If a source is emitting the radiation having the electric field in two mutually perpendicular plane simultaneously then such radiation is termed as randomly polarized light. Or in other words if both, s polarized and p polarized wave, are present simultaneously the wave is randomly polarized light .
Linearly or Plane polarized light:
Let us consider a plane electromagnetic propagating along z axis whose electric field can be expressed by
............................(1)
In this case, the electric field will be sinusoidal as a function of time for a given location,z, in the path of the propagation or for a given time, the electric field will go sinusodially as function of z as shown in figure1. But the direction of electric field at any time and at any point in space is always same i.e along the x-axis for the representation of eq 1.
Similarly we can have a have an plane wave propagating along the z-axis having the electric field along the y axis given by
.............................. (2)
There are two differences between the wave represented by eq.(2) with that of eq.(1). There is a phase difference 
between the two waves and the direction of the electric filed is always along y axis for eq.(2).
fig..(1)
Suppose that the frame of reference is rotated about z axis by an angle 
in anticlockwise direction as shown in fig.2
Let us denote the new corresponding axis as x' and y' and their unit vectors as 
and 
respectively. In this new frame of reference the electric field associated with the electromagnetic wave represented by by eq 1.
or,
..........(3)
where 
represents the component of the electric field along x' and 
that of along y' for the eq 3 (which is representing the same field of eq 1).