It is important to realize that the darkness seen after rotating the second polarizer is due to interference of light after the beams have passed through the second polarizer.
The third polarizer can be orientated such that light at \(180^o\) and \(90^o\) phase are diminished, the result is a brighter beam due to light at phase \(0^o\) and \(270^o\) that does not cancel when they interfere.
A polarizer produce a \(0^o\) phase and a \(90^o\) phase beam. A second polarizer can be rotated in front of the first to block \(0^o\) phase and allow \(90^o\) phase, or block \(90^o\) phase and allow \(0^o\) phase or block both phase by causing them to interfere after the polarizer.
A \(90^o\) phase beam can produce a diminished \(90^o\) phase beam and a brighter \(180^o\) phase beam after a polarizer. At the same time, a \(0^o\) phase beam can produce a diminished \(90^o\) phase beam and a brighter \(0^o\) phase beam after the same polarizer.
This is the case, after second and third polarizer. After the second polarizer is darkness as the beams interfere destructively. The third polarizer results in two beams that do not cancel.
This one is nutty...
