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If you simply take two polarizers (the old fashond pre-autofocus kind or simple polarizing media, at least for the front one) and use an angle between them, you get the effects of polarizing filters, not just dark glasses. Am I right in assuming that a variable ND filter doesn't introduce these effects? If so, just how does it work?

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No, a variable ND filter is just two polarisers, the first one being linear and the second being either linear or circular. Thus when using a variable ND you also get the effects of a polariser.

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    This is not true! Variable ND filters can not be used for CP effects. See my answer. – doug Oct 21 '17 at 6:00
  • @scottbb Agree re current ND filters. I was referring to Matt's answer re two polarized filters. I should have been clearer. These days ND filters are circularly polarized on entry and exit and are reasonably effective, variable density ND filters w/o polarization effects such as one might want with CP filters. – doug Apr 8 at 1:50
  • @scottbb Sorry, standard construction of variable ND filters is a QWP followed by a linear polarizer and the portion rotating relative to the first two elements also consists of a linear polarizer followed by a QWP. The entry QWP keeps the VND filter from exhibiting polarized filter effects like darkened blue skys and attenuating reflections off of water. The X-Effect is intrinsic to the two inner polarized filters and the angle light goes through them. It's an intrinsic problem with VND filters operating at high attenuation. Just compare a VND to a CP which does exhibit polarization effects – doug Apr 8 at 2:40
  • @scottbb If there were no QWP in front of the first LP, then the variable ND would exhibit standard polarized filter effects just like a standard CP. They don't. The first QWP is not useless. It circularly polarizes light that is already polarized. This is why variable NDs don't attenuate reflections off of water. Var NDs are really just two CP filters with the LP facing each other and one able to rotate relative to the other. – doug Apr 8 at 6:22
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    @scottbb No problem. I appreciate strong pushback. It makes me more careful and think through answers more carefully. That people strongly engage w/o going nuts (usually) is one of the valuable things here. And my initial comment was not as clear as it should have been. – doug Apr 8 at 22:46
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You are correct. Variable ND filters do not introduce the polarization effects that CP filters do and do not increase sky blue saturation or reduce water reflections.

They are composed of 4 layers constituting 2 circular polarizing filters. Each filter is composed of a 1/4 wave plate, which converts linear polarization to circular polarization on the outer surfaces and linear polarization layers on the inner surfaces. The 1/4 wave plate on both outer surfaces eliminates the selective attenuation of linear polarization in scenes, as well as preventing the adverse effects of linearly polarized light on an SLR's exposure sensor.

However, the X (cross effects) are still there to some degree and are worse the darker the variable ND is set and the wider the camera's aperture.

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    What you are calling 'circular polarizers' aren't polarizers at all, they're quarter-wave plates. An actual circular polarizer is a linear polarizer with a quarter-wave plate behind it. – Michael C Jun 15 '18 at 14:53
  • @MichaelClark Yes, the 1/4 wave plate does not, in itself produce circularly polarized light. A quarter wavelength plate only converts linearly polarized light to circularly polarized light. A linear polarizer has to be in front to produce circularly polarized light from non polarized light. I'll edit for clarity. – doug Jun 15 '18 at 17:49
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"Unfortunately" the ND filter is made of two polarisers. The simplest case is when you use two linear polarisers and depending on the angle between them, different amount of light will be removed.

Nowadays one linear filter is substituted with a circular one, due to problems with the AF system. In that case CPL filter is screwed at the top of the linear one.

Usage of polarising filters is obviously introducing problems with a "X effect" at some position.

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    The circularizer is last, that is closest to the lens. If it was on top the circular light would be blocked by a constant amount (50% for ideal linear polarizer) regardless of the orientation. – JDługosz May 22 '15 at 14:21
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Doug is correct — to remove any polariser effect the light is 1: circularly polarised; then 2: Two linear polarisers provide the attenuation (depending on the angle between them); then 3: a final circular polariser 'de-polarises' the light to avoid autofocus problems. If you had manual focus only and wanted polariser effects you could remove both circular polarisers leaving only the linear polarisers — but it would not be easy to align them for the desired polarisation effect AND get the attenuation you want..

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    What you are calling 'circular polarizers' aren't polarizers at all, they're quarter-wave plates. An actual circular polarizer is a linear polarizer with a quarter-wave plate behind it. – Michael C Jun 15 '18 at 14:45

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