Question involving Polarization and Quarter Wave Plates?

Question

Unpolarized light, which travels in the z-direction and has frequency f0, enters a linear polarizer with a Transmission Axis (TA) inclined at an angle of 45 degrees with respect to the x-axis. It then enters a birefringent slab and emerges as circularly polarized light. The TA and the fast and slow axes of the birefrigent slab are parallel to the xy plane

With the slab oriented as shown in the figure, the light frequency is doubled to 2 f0. After the light emerges from the birefringent slab, it is:
a) unpolarized
b) circularly polarized
c) linearly polarized

the answer is c) but I cant figure it out for the life of me, can anyone explain?

Answer 1

If you pass the polarized light through a quarter-wave plate, the wave comes out circularly polarized. When you double the light frequency, that plate is no longer a quarter-wave plate, it is a half-wave plate. A half wave plate merely rotates the plane of polarization of an incoming polarized wave.

Answer 2

There seems to be something wrong with the question. Title contains the mention of quarter wave plate but in the details only bi-birefringent slab is mentioned. If unpolished light enters any birefringent plate then it splits into two components. If they are made to travel along the same direction then phase difference would be created between two vibrations giving rise to different kinds of polarizations but only if the incident light is polarized. So there seems to be some confusion in the question about the sequence of what after what or about whether it is a quarter wave plate or half wave plate. Of course I have never heard about the change of frequency. That seems to be utterly impossible.