Amplitude of unpolarized EM wave after a polarizer

The amplitude of a linearly polarized wave after a polarizer is $$E_1 = E_0\cos(\theta)$$

and the intensity is

$$I_1 = I_0\cos^2(\theta)$$

Now for unpolarized light, the time averaged amplitude is zero and the time averaged intensity is

$$\langle I_1\rangle_t = \langle I_0\cos^2(\theta)\rangle_t = \frac{I_0}{2}$$

But what is the amplitude of the field after the polarizer ?

If i say $\langle I_1\rangle_t = I_1 = \frac{1}{2}|\tilde{E}^2| = \frac{1}{2} \tilde{E}\tilde{E}^* = \frac{E_1^2}{2}$ , then $E_1 = \sqrt{I_0} = E_0$

but it can't be right... what have I missed ?

• Hint: can you have coherent, monochromatic, unpolarized light? – Emilio Pisanty Apr 26 '16 at 23:11
• Related: Electromagnetic field of unpolarized light and similar questions on this site. – Emilio Pisanty Apr 27 '16 at 0:00
• My book says that unpolarized light is equivalent on average to the superposition of two incoherent orthogonal linear polarizations. It can be monochromatic though. But I don't see how to get the field after the polarizer from this. If the incoming wave' s phase and polarization are random then after the polarizer it would be something like $E_1 = E_0(t)\cos(\theta(t))$ wouldn't it ? – mwa1 Apr 27 '16 at 13:21
• Does a field like $\cos(\theta(t))$ seem particularly monochromatic? Seriously, though - unpolarized light can only be incoherent, and it won't get coherent simply by projecting out one polarization. Asking for the electric field amplitude of incoherent light doesn't make that much sense, to be honest, at least not without a deep understanding of how your light is incoherent and a much more sophisticated view of the electric field's oscillations. – Emilio Pisanty Apr 27 '16 at 16:29
• Well, I'm only asking that because it's in my homework on the Lyot filter... "Unpolarized monochromatic wave with ampitude $E_0$ travels along the z axis and passes through polarizer a P_1 and a waveplate W_1-> Write the complex expression of the electric field coming out of the P_1 on the axes of the neutral lines of W_1" Did I misunderstand something ? – mwa1 Apr 27 '16 at 16:49