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I can buy a Vivitar brand polarizing filter from Walmart for less than $13, or I can buy a different brand from a photo store in the range of $60 to up over a $100. Instinct tells me that the Vivitar filter will not be as good, but why? Does it have to do with coating or is there another reason? Why should I avoid using the Vivitar brand filter? They seem like a reputable company, no?

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A filter may be not neutral (either at all or with respect to certain "filter vs light polarization plane" orientations) - that is, white balance will be shifted sideways.

A filter may be not coated - that is, the image contrast will be lower in certain conditions.

A filter may be not circular - that is, may cause problems with AF of specific cameras (and this is not the case - the filter which you gave link to is linear+circular).

However, I am using some cheap polarizer for casual photographs and I did not regret it even once, even with sun shining inside frame.

I have also checked the claim about light transmission from this answer. My cheap "Green.L" filter (not even coated) looses 1,32 stop of light. It is almost same as Hoya HD (which looses same amount of light judging by some reviews). It may be different with other models though.

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