Why does the intensity of unpolarized light reduce to half after passing it through a polarizer?

Question

When unpolarized light is polarized with two polarizers, the intensity becomes $I=I_0\cos^2(θ)$ (Malus's law). But when unpolarized light is polarized with only one polarizer, the intensity is reduced to half the intensity of the unpolarized light. Why?

Answer 1

Malus's law is about the effect of a polariser on polarised light. You've clearly read a badly written version of it. What your author likely meant to say was:

1. One begins with unpolarised light;

2. The first polariser quells the unaligned component of the unpolarised light and outputs polarised light (with half the input's intensity). This polarised output has intensity $I_0$ in your notation;

3. Of the polarised output from the first polariser, the second polariser lets through a fraction $(\cos\theta)^2$ where $\theta$ is the angle between the axes of the polarisers. So I say again: $I_0$ is the intensity of the polarised input to the second polariser, not the intensity of the unpolarised input to the system of two polarisers. With this proviso, the output intensity is $I_0\,(\cos\theta)^2$.

In Answer to:

But I don't understand why the intensity is lowered to half the input's intensity after the first polariser?

Depolarised light is actually quite a subtle and tricky concept: I discuss ways of dealing with it in my answers here and here. You can think of it intuitively: without a preference for polarisation, perfectly depolarised light must dump half its energy into a polariser: you can take this as a kind of "definition" of depolarised light if you like. The quantum description is much simpler than the classical, so I reproduce it here. We imagine the source producing photons each in pure polarisation states but with random direction. That is, each photon's polarisation axis makes some random, uniformly distributed angle with the polariser's axis. Its pobability of absorption is therefore $(\cos\theta)^2$. So now we simply average this quantity given a uniformly distributed angle:

$$\left<(\cos\theta)^2\right> = \frac{1}{2\pi}\int_0^{2\pi} (\cos\theta)^2\,\mathrm{d}\theta = \frac{1}{2}$$

to find the overall probability of passage through the polariser, and thus the proportion of photons that make it through. For more info on what polarisation actually means for a single photon, see my answer here.