I have a throlabs half- and quarter-wave plate with rotation plate. There shown the angle scales and a line denotes the fast axis. But what does the angle mean? I do some research and someone said the angle reading tells the angle away from the transmission direction and other said that's the angle away from the fast axis. So here is the transmission direction same as the fast axis?

If I know the incoming light is linear polarized but don't know the orientation. Is that possible to tell the orientation by using the half-wave plate?

Also, I am studying the same optics element. When I read the material of the polarized beam splitter, I know that the incident beam will be split by the splitter with the two perpendicular outgoing beams.

What really confusing is if the incident light is linear polarized, what can I tell about the outgoing beams after a splitter?

Can I say they are bother linear polarized and what about the polarized orientation?

The last question is pretty odd to me. In the text, it said we can use a quarter-wave plate to elliptical polarize a linear-polarized light. In some special case, the outgoing light could be circularly polarized. But how can I verify that? I tried the following: I let the linear-polarized light go through a half-wave plate so to control the orientation of the polarization, then let that light pass a quarter-wave plate. If I measure the power out of the quarter-wave plate, it is pretty constant. I think it doesn't tell if the beam is elliptical or circular because the power I measure is the average one, right? So I put a polarized beam splitter after the quarter-wave plate to observe the power of the split light. I think if the light is elliptical, the power should change with time, but again it is pretty constant. Doesn't matter how I rotate the quarter-wave or half-wave plate, the power is pretty constant.

Is that anything wrong with this testing way?

Have a look at this Wikipedia page, it explains the difference between a fast axis and a slow axis. But essentially it is to do with the properties of a birefringent crystal.

Firstly, you could consult the Thor Labs manual/booklet that you have got with the equipment, that should sort out your confusion for you. I'm guessing it is the angle away from the fast axis. But since (as far as I remember) the angular scale can be detached and put back onto the polarizer, it will help if you calibrate your polarizer with some known light before you make any measurements with it.

As daaxix has pointed out, you could use Stoke's parameters to characterize the light. There are several standard experiments in the literature to determine the Stoke's parameters.

If you send a linearly polarized beam through a quarter wave plate, it will in general come out elliptically polarized. You could confirm this by putting another polarizer (with a known polarization axis) after your first polarizer and rotate that with respect to the first polarizer, and then measure the intensity. The resulting intensity will be constant with angle if your light is circularly polarized, or it will vary (the variation will be larger than your experimental error) if the light is elliptically polarized.

You can actually use a linear polarizer and a quarter waveplate to measure the Stokes vector of the incoming light. This will give you a robust measure (to second order statistically) of the polarization of the incoming light.

You will need to have the fast axis of the waveplate aligned to the axis of the polarizer in order to get a good measurement.

For how polarimeters work, look up Azzam, Chipman, or Collett. A good fast and uncomplicated basic polarization reference is Collett's SPIE Field Guide on Polarization.

About how the splitter affects the polarization, well they teach an entire class about such things here at the Optical Sciences College, so the answer is it depends on what kind of splitter you have. You can compute what the splitter is doing, but you need to know all the coating data, all the angles of incidence of the incoming light, etc.

As to your last question, which I understand to be asking "how to measure which elliptical state that I am generating" you could use the following procedure :

• Set up two crossed polarizers. Minimize the power on the detector to do so.
• Insert the quarter waveplate in between them. Align the fast axis (should be marked) to the axis of the first polarizer approximately. Rotate the quarter waveplate until the power is again minimized. This sets your fast axis and first polarizer axis parallel with each other.
• Next, rotate the polarizer (hopefully you have a micrometer or
something on the rotation stage) to known angles and measure the
power. The power should change and give you something that looks
like two waves added together (a carrier frequency and envelope).
Then you can fit the parameters of the Mueller matrix for a linear
retarder to that curve. The Mueller matrix is described here in Chipman's chapter on Mueller matrices in The Handbook of Optics II.
• Once you have the parameters of the Mueller matrix, you can compute the outgoing Stokes vector from the linear polarizer followed by the quarter waveplate, based on the angle of rotation. You can convert the Stokes vector to an ellipse if you want...