Half Wave-Plate: measurement contradicting theory

We are measuring our custom-made Half-Wave-Plates (HWPs) for controlling retardation error as part of incoming QC. We don't have a device for measuring retardation nor do we have access to X-ray analyzer.

We have experimental result with no model supporting it; the actual question is in bold below. Background in italics. I apologize for the long story but think it is needed for the helpful answer.

As a practical guideline, we know that a Half Wave Plate (=HWP) rotates the polarization by twice the angle to the optical axis, so HWP at 45° between optical axis and mechanical reference will rotate the polarization by 90°. Our HWP is "Zero Order Waveplate" which means the HWP is composed of two Birefringent plates with their optical axis rotated 90°, and their thickness differ by a precise amount for which the light phase changes (retards) by 180°. But a real HWP will not be perfect and its retardation will not be exactly 180°. The difference is the retardation error.​

Years ago, the method was simply a setup with a laser and two parallel polarizes. Measure the power without the HWP and measure the power when the HWP is dropped on a mechanical reference between the polarizers. By calculating the ratio (extinction) and rejecting HWPs with too high values, we thought we maintain retardation error.
The extinction is T||=1- sin22θ*sin2Φ/2 with θ being the angle between the optical axis and the mechanical reference, Φ is the retardation; this formula can be proved by Jones matrix of rotated retarder in between parallel polarizers.
I now think this can keep the combined retardation error (thickness error and optical axis to mechanical axis angle error) at bay, but it is not practical when trying to tighten the tolerances.

Defining the axis & terminology: Z is the optical axis (laser propagates along Z). Y is the up/down and X the left/right. "Flipped" means rotation around Y, θy=180°, so the HWP front surface, that was facing the laser now becomes the back surface and faces the detector. θy=0° and θy=180° are called "orientations".
Define minimum extinction as the minimum extinction when minimizing over θz.​

It was noted that sometimes the measured extinction vary significantly if the HWP is flipped.
My theory was that the angle between the optical axis and the mechanical reference was not exactly 45° so when flipping, the system is not symmetric therefore the extinction vary between the two orientations. It can be shown that the difference in retardation between orientations will be 180°+4ε, ε being the error in the optical-to-mechanical axis, regardless of jig misalignment.

The solution was to improve the stage holding the HWP so we can adjust θz for minimum extinction, compensating for any error in θ and measuring the effect of Φ only, directly correlated to the retardation error.
And here comes the question: Some of the HWPs (but not all) have significantly different minimum extinctions when flipped. I cannot find a theoretical explanation nor can I vary the Zemax models I have to explain the experiment but it is repeatable.Have you encountered such an odd behavior? Any ideas what can cause it?

With the phase retarder at 45deg, the polarization at the interface between the two birefringent materials will be elliptical, i.e. partly linear and partly circular.

If the glue or however the two materials are attached together is somehow has different transmission for one polarization (left vs. right or linear vertical vs linear horizontal), then flipping the filter will change the optical properties.

Otherwise you could try setting up a model of Jones matrices for all the elements. The main problem there is figuring out what happens to the matrix of the HWP when you flip it.

Thanks M Quack and mfb,
I tried modeling the input polarizer and the analyzer angle misalignment, together with random errors of the HWP, using Zemax. Still got the same minimum extinction when flipped.
I was thinking of reflection after a colleague of mine suggested it on Friday. This means the is either reflecting differently or absorbing differently the two orthogonal polarization directions. As per reflection, the angle of incidence is 0 so normally any AR coating will behave exactly the same for both polarization directions (as far as I remember, s-pol and p-pol are always identical for small angles). The only way I can imagine a difference that will affect reading when flipped is if the coating is actually polarizing. Something like Polaroid. I know the AR coating is an Oxide coating and done using Ion Assisted Deposition. Does anybody know of such a coating becoming slightly polarizing?
M Quack, this idea that something happens at the interface, and it becomes polarizing, is interesting. The problem happens both with glue and with optical contact. Any idea how an optical contact can become a polarizer? Any documentation of this mechanism I can use to further research?

It is possible that you shear the glue during cementing, that might lead to a similar effect. This should also be observable if you glue two plates of glass using the exact same procedure.

Another possibility is that glue or coatings orient due to interaction with the birefringent material - to be optically anisotropic it has to be structurally anisotropic, which might lead to molecules in the glue/coating lining up with the optical axes. As this is due to the anisotropy of the optical material it cannot be tested with glass.

As long as the beam is perfectly perpendicular to the surface there is no difference between the s- and p-polarizations. As soon as the alignment is less than perfect there will be a small difference.

I assume that you have verified that the entrance and exit surfaces of your HWP are nicely parallel? In that case flipping should not make a difference, unless there are alignment errors (which should not be reproducible).

I assume that you have verified that the entrance and exit surfaces of your HWP are nicely parallel? In that case flipping should not make a difference, unless there are alignment errors (which should not be reproducible).

Thanks!
I am confident the plates are parallel (have no wedge). I used a camera instead of the sensor, and the bit of structure I see there comes from my source (removing the HWP does not change the structure, just the intensity). Also the HWP is perpendicular to the beam, by checking the back reflection.

I don't think the root-cause is the glue, as it happens also with "optical contact" which means no glue at all in the beam path.

I don't know enough about coating behavior on anisotropic material. It sounds feasible that the coating molecules will follow the Quartz molecular structure. I cannot explain to myself why would the two sides of the HWP (both sides are AR coated) behave differently, but let's assume this is the case. How can I empirically prove it?
Would it be enough to see a difference between orientations without the 2nd polarizer (aka the analyzer)?

I would assume that all your HWP have the same AR coating on both sides, but just in case they don't, I've seen cases where feedback (reflections) going back into the laser cavity can cause small but consistent changes in the power output.

Is it possible to remove the surface coating from a "bad" HWP that would be rejected anyways?

Thanks. It would certainly be helpful to remove the coating from a "bad" HWP and see if it behaves as expected without the coating, or not. Worth a try. I asked our vendor, but will have an answer only in two days.

Thanks. It would certainly be helpful to remove the coating from a "bad" HWP and see if it behaves as expected without the coating, or not. Worth a try. I asked our vendor, but will have an answer only in two days.

(almost) final answer is no, we cannot remove the AR coating without ruining the waveplate.
I am still interested in gathering this forum's experience to help figuring out the mystery of the flipped HWP.