Unpolarized light enters the polarizer and gets polarized at one certain angle. If we place an analyzer behind the polarizer and align them, we observe that all light is transmitted. If we rotate the analyzer by an angle $\theta$, then the intensity of light leaving the analyzer would fall as $\cos^2(\theta)$. What I am asking is: is there some device that when placed instead of the analyzer will transmit light as a delta function: max intensity for $\theta = 0$ (perfect alignment) and zero intensity for all other angles ($\theta$ not equal to zero)?
Clarification: user1800 was already referring to a 2nd polarizer when he used the word analyzer, so his question is legitimate.
I don't think there is such a device in classical optics.
But one can make the response sharper than the $cos^2(\theta)$ when lasing thresholds and nonlinear effects are taken into account.
A sharper response happens e.g. inside a high Q laser cavity (e.g. HeNe, Ar+). Brewster windows have very low losses only for the 'exactly' right polarization, giving those modes a competitive advantage.
If you had pulses, you could combine the polarizer/analyzer combo with a nonlinear device (e.g. saturable absorber) to discriminate against low pulse powers. Such effects are weak, and only work well inside laser cavities with high Q (many roundtrips, mode competition), see e.g. in the CPM dye lasers of the 80s.
If you can't be inside the cavity with your analyzer, you can finally use an external, coupled cavity. Think of a pulsed laser that can only overcome threshold if an outcoupled pulse is coupled back with the right timing relative to the internally circulating pulse (or rather ASE, since it isn't a pulse yet).
This effect can also be used in the reverse. A friend once set up his fs dye laser, such that it turned on when one entered the room (no electronics). A 2nd external cavity had an outcoupled pulse timed such that it would deplete the gain (in the presense of a saturable absorber elsewhere in the cavity) too early and suppress lasing. If one interrupted that cavity by walking through the invisible, sub-threshold beam, the laser would magically start in the main cavity.